There are a number of drugs, not just Lantus and Lev, that are depot medications. The term is from pharmacology and essentially means that a portion of the active drug is stored and gradually absorbed and used. Some depot drugs are very long acting -- they can last for weeks. In the case of insulin, the depot allows for a longer duration.
The way Lantus works, you inject the insulin and because it is slightly acidic and you're injecting into an environment with a different pH, it forms micro crystals. These crystals deposit in the fat tissue under the skin. These crystals dissolve which is why Lantus has a longer duration.
Because you are injecting more insulin every 12 hours and not all of the insulin that has previously been injected is used, there is a "pool" of insulin that your kitty is drawing upon. So, when you increase the dose, it takes a few days for the depot to catch up to the new dose. When you reduce the dose, the depot may be overly full and is likewise can take a few days for the excess insulin to be used up.